= Solution
The <characteristic function> of $B(t)\sim N(0,t)$ gives $\mathbb E e^{2iB(t)}=e^{-2t}$. Taking real parts and using $\sin^2b=(1-\cos2b)/2$,
$$
\boxed{\mathbb E\Delta(t)=\frac{1-e^{-2t}}2,
\qquad \lim_{t\to\infty}\mathbb E\Delta(t)=\frac12.}
$$
There is also a direct check from the <stochastic differential equation>. Its <Itô integral> has zero <expectation> because its integrand is bounded. Thus $m(t)=\mathbb E\Delta(t)$ satisfies $m'(t)=1-2m(t)$ and $m(0)=0$, whose solution is the same expression.
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